Examples

The case \(p=1\) is called the trace norm or nuclear norm, where we find \[d_i ~=~ |l_i| \sum_j |l_j|
\qquad
\text{and}
\qquad
V ~=~ {\Big(\sum_j |l_j|\Big)}^{2}
.\] The case \(p=2\) is called the Frobenius norm where we find \[d_i ~=~ l_i^{\frac{2}{3}} \sum_j l_j^{\frac{4}{3}}
\qquad
\text{and}
\qquad
V ~=~ {\bigg(\sum_j l_j^{\frac{4}{3}}\bigg)}^{3/2}
.\] The case \(p=\infty\) (maximum eigenvalue) is called the spectral norm where we find \[d_i
~=~
\sum_j l_j^2
\qquad
\text{and}
\qquad
V
~=~
\sum_j l_j^2
.\] Note that in this last case the \(d_i\) are all the same, so the optimal matrix is a multiple of the identity matrix.

A case of interest is also \(p=0\). The above analysis probably does not work (as \(p\)-norms switch from concave to convex at \(p=1\)), but we can still show in a different way that the final formula gives the correct answer, i.e. we find \[d_i
~=~
n
l_i^2
\qquad
\text{and}
\qquad
V
~=~
n
.\]